A different kind of arithmetic
A mathematical preprint reports an affirmative answer to Tarski's general finite-coefficient sum problem for linear orders. The result is the central outcome of a broader effort to build an arithmetic for ordered sums, the operation that places one linear order after another.
The paper studies the class LO of linear orders, arrangements in which any two elements can be compared, and asks what structural rules govern their ordered sums. Rather than assigning numbers to the orders, it compares their shapes through embeddings, isomorphisms and repeated sums. The goal is to determine when different arrangements are actually the same order type, and when a relationship between two sums forces a relationship between their component orders.
The route through division
The main tool is a Euclidean algorithm for pairs of linear orders that satisfy one-sided weak commutativity. The paper develops left and right versions. At each stage, the comparison asks whether one ordered sum can be embedded onto an initial segment or a final segment of the other. In that way, a procedure familiar from ordinary division becomes a structural test for ordered objects.
Using that framework, the authors report unified generalizations and proofs of classical results for ordered sums, including Lindenbaum's division theorem and an Aronszajn representation theorem for additively commuting pairs. The generalized division result says that if positive finite multiples of two orders are isomorphic, there is a common order that makes the original orders corresponding reduced multiples.
Rules for commuting sums
One theorem gives the headline answer to Tarski's general finite-coefficient problem: for linear orders, the paper reports an affirmative result. Another addresses offset sums made from two orders. If the relevant sums have the same order structure, the component orders commute under ordered sum. In plain terms, the order in which the two components are placed does not change the resulting order type in that case.
For pairs of linear orders more generally, the paper gives an exact characterization of commutation through two possible directional patterns. One pattern requires the right-infinite repetition of A to fit into the beginning of the corresponding repetition of B, while the left-infinite repetition of A fits into the end of B. The other pattern reverses A and B. Here, right-infinite and left-infinite refer to repeated sums extending in opposite directions.
A second characterization describes the same pairs arithmetically. The combined sum must either be isomorphic to one operand, so that the sum is absorbed into A or B, or the right-infinite sums of the two orders must match and their left-infinite sums must match as well. The two descriptions give different views of the same result: one follows how orders embed at their boundaries, and the other follows what repeated addition does to them.
A classification beyond pairs
The paper then broadens the question from pairs of orders to commutative semigroups representable inside ordered sums. Its classification is exact: the representable semigroups are precisely the stated replacement semigroups. These are organized over a linear order, with each position carrying either a subsemigroup of the positive cone of a strict real subgroup or a one-element semigroup. The result therefore supplies both a construction and a test for membership in the class described by the paper.
A further construction concerns a non-terminating left division system. It represents the two-sided infinite product of the initial order by replacing the real line with orders indexed by split translation-orbit classes. The construction connects the division process to the replacement structures used elsewhere in the paper.
The scope of the result
All of these findings are about linear orders, ordered sums and the associated semigroup representations. The supplied record identifies the work as an arXiv preprint. Its claims are structural results within that mathematical setting, rather than measurements of an external population or process.
Paper data and sources
Original title: The Additive Arithmetic of Linear Orders
Authors: Garrett Ervin, Eric Paul
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text